Immunising two liabilities with a single zero-coupon bond

Question

An investment fund has liabilities of £11 million due in 7 years’ time and £8.084 million in 11 years’ time.

The manager of the fund will meet the liabilities by investing in zero-coupon bonds.

The manager is able to buy zero-coupon bonds for whatever term is required and there are adequate funds at the manager’s disposal.

(i) Explain whether it is possible for the manager to immunise the fund against small changes in the rate of interest by purchasing a single zero-coupon bond.

This question from an actuarial exam carries only 2 points, so I assume it must be simple, but I don't know how to solve it.

The only idea I have is to check whether a single zero-coupon bond would meet conditions for Redington immunisation.

However, not knowing the value or term of the bond, or the interest rate, this would be complicated or impossible to calculate.

Thank you in advance for any help.

Answer 1

The exercise as stated is not really clear, I agree. So here is one interpretation which makes sense to me: You can't immunise with one bond because there are two maturities. You could buy a zero-bond matching the duration of the combined liability cash-flows but its maturity would fall somewhere between the two maturities of your liability. Hence once your duration matched zero bond redeems you would be exposed to the longer liability.

Answer 2

No, as with only a single asset, the spread of the asset proceeds would be less than the spread of the liability outgo (at times 7 and 11). Thus, the convexity of the assets would be less than the convexity of the liabilities and the third condition of immunisation (Redington immunization) could not be satisfied.